diff -r f04b33ce250f -r a4dc62a46ee4 Integ/Equiv.ML --- a/Integ/Equiv.ML Tue Oct 24 14:59:17 1995 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,272 +0,0 @@ -(* Title: Equiv.ML - ID: $Id$ - Authors: Riccardo Mattolini, Dip. Sistemi e Informatica - Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1994 Universita' di Firenze - Copyright 1993 University of Cambridge - -Equivalence relations in HOL Set Theory -*) - -open Equiv; - -(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***) - -(** first half: equiv(A,r) ==> converse(r) O r = r **) - -goalw Equiv.thy [trans_def,sym_def,converse_def] - "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r"; -by (fast_tac (comp_cs addSEs [converseD]) 1); -qed "sym_trans_comp_subset"; - -goalw Equiv.thy [refl_def] - "!!A r. refl(A,r) ==> r <= converse(r) O r"; -by (fast_tac (rel_cs addIs [compI]) 1); -qed "refl_comp_subset"; - -goalw Equiv.thy [equiv_def] - "!!A r. equiv(A,r) ==> converse(r) O r = r"; -by (rtac equalityI 1); -by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1 - ORELSE etac conjE 1)); -qed "equiv_comp_eq"; - -(*second half*) -goalw Equiv.thy [equiv_def,refl_def,sym_def,trans_def] - "!!A r. [| converse(r) O r = r; Domain(r) = A |] ==> equiv(A,r)"; -by (etac equalityE 1); -by (subgoal_tac "ALL x y. : r --> : r" 1); -by (safe_tac set_cs); -by (fast_tac (set_cs addSIs [converseI] addIs [compI]) 3); -by (ALLGOALS (fast_tac (rel_cs addIs [compI] addSEs [compE]))); -qed "comp_equivI"; - -(** Equivalence classes **) - -(*Lemma for the next result*) -goalw Equiv.thy [equiv_def,trans_def,sym_def] - "!!A r. [| equiv(A,r); : r |] ==> r^^{a} <= r^^{b}"; -by (safe_tac rel_cs); -by (rtac ImageI 1); -by (fast_tac rel_cs 2); -by (fast_tac rel_cs 1); -qed "equiv_class_subset"; - -goal Equiv.thy "!!A r. [| equiv(A,r); : r |] ==> r^^{a} = r^^{b}"; -by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1)); -by (rewrite_goals_tac [equiv_def,sym_def]); -by (fast_tac rel_cs 1); -qed "equiv_class_eq"; - -val prems = goalw Equiv.thy [equiv_def,refl_def] - "[| equiv(A,r); a: A |] ==> a: r^^{a}"; -by (cut_facts_tac prems 1); -by (fast_tac rel_cs 1); -qed "equiv_class_self"; - -(*Lemma for the next result*) -goalw Equiv.thy [equiv_def,refl_def] - "!!A r. [| equiv(A,r); r^^{b} <= r^^{a}; b: A |] ==> : r"; -by (fast_tac rel_cs 1); -qed "subset_equiv_class"; - -val prems = goal Equiv.thy - "[| r^^{a} = r^^{b}; equiv(A,r); b: A |] ==> : r"; -by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1)); -qed "eq_equiv_class"; - -(*thus r^^{a} = r^^{b} as well*) -goalw Equiv.thy [equiv_def,trans_def,sym_def] - "!!A r. [| equiv(A,r); x: (r^^{a} Int r^^{b}) |] ==> : r"; -by (fast_tac rel_cs 1); -qed "equiv_class_nondisjoint"; - -val [major] = goalw Equiv.thy [equiv_def,refl_def] - "equiv(A,r) ==> r <= Sigma(A,%x.A)"; -by (rtac (major RS conjunct1 RS conjunct1) 1); -qed "equiv_type"; - -goal Equiv.thy - "!!A r. equiv(A,r) ==> (: r) = (r^^{x} = r^^{y} & x:A & y:A)"; -by (safe_tac rel_cs); -by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1)); -by ((rtac eq_equiv_class 3) THEN - (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3)); -by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN - (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1)); -by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN - (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1)); -qed "equiv_class_eq_iff"; - -goal Equiv.thy - "!!A r. [| equiv(A,r); x: A; y: A |] ==> (r^^{x} = r^^{y}) = (: r)"; -by (safe_tac rel_cs); -by ((rtac eq_equiv_class 1) THEN - (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1)); -by ((rtac equiv_class_eq 1) THEN - (assume_tac 1) THEN (assume_tac 1)); -qed "eq_equiv_class_iff"; - -(*** Quotients ***) - -(** Introduction/elimination rules -- needed? **) - -val prems = goalw Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r"; -by (rtac UN_I 1); -by (resolve_tac prems 1); -by (rtac singletonI 1); -qed "quotientI"; - -val [major,minor] = goalw Equiv.thy [quotient_def] - "[| X:(A/r); !!x. [| X = r^^{x}; x:A |] ==> P |] \ -\ ==> P"; -by (resolve_tac [major RS UN_E] 1); -by (rtac minor 1); -by (assume_tac 2); -by (fast_tac rel_cs 1); -qed "quotientE"; - -(** Not needed by Theory Integ --> bypassed **) -(**goalw Equiv.thy [equiv_def,refl_def,quotient_def] - "!!A r. equiv(A,r) ==> Union(A/r) = A"; -by (fast_tac eq_cs 1); -qed "Union_quotient"; -**) - -(** Not needed by Theory Integ --> bypassed **) -(*goalw Equiv.thy [quotient_def] - "!!A r. [| equiv(A,r); X: A/r; Y: A/r |] ==> X=Y | (X Int Y <= 0)"; -by (safe_tac (ZF_cs addSIs [equiv_class_eq])); -by (assume_tac 1); -by (rewrite_goals_tac [equiv_def,trans_def,sym_def]); -by (fast_tac ZF_cs 1); -qed "quotient_disj"; -**) - -(**** Defining unary operations upon equivalence classes ****) - -(* theorem needed to prove UN_equiv_class *) -goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)"; -by (fast_tac (eq_cs addSEs [equalityE]) 1); -qed "UN_singleton_lemma"; -val UN_singleton = ballI RSN (2,UN_singleton_lemma); - - -(** These proofs really require as local premises - equiv(A,r); congruent(r,b) -**) - -(*Conversion rule*) -val prems as [equivA,bcong,_] = goal Equiv.thy - "[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)"; -by (cut_facts_tac prems 1); -by (rtac UN_singleton 1); -by (rtac equiv_class_self 1); -by (assume_tac 1); -by (assume_tac 1); -by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]); -by (fast_tac rel_cs 1); -qed "UN_equiv_class"; - -(*Resolve th against the "local" premises*) -val localize = RSLIST [equivA,bcong]; - -(*type checking of UN x:r``{a}. b(x) *) -val _::_::prems = goalw Equiv.thy [quotient_def] - "[| equiv(A,r); congruent(r,b); X: A/r; \ -\ !!x. x : A ==> b(x) : B |] \ -\ ==> (UN x:X. b(x)) : B"; -by (cut_facts_tac prems 1); -by (safe_tac rel_cs); -by (rtac (localize UN_equiv_class RS ssubst) 1); -by (REPEAT (ares_tac prems 1)); -qed "UN_equiv_class_type"; - -(*Sufficient conditions for injectiveness. Could weaken premises! - major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B -*) -val _::_::prems = goalw Equiv.thy [quotient_def] - "[| equiv(A,r); congruent(r,b); \ -\ (UN x:X. b(x))=(UN y:Y. b(y)); X: A/r; Y: A/r; \ -\ !!x y. [| x:A; y:A; b(x)=b(y) |] ==> :r |] \ -\ ==> X=Y"; -by (cut_facts_tac prems 1); -by (safe_tac rel_cs); -by (rtac (equivA RS equiv_class_eq) 1); -by (REPEAT (ares_tac prems 1)); -by (etac box_equals 1); -by (REPEAT (ares_tac [localize UN_equiv_class] 1)); -qed "UN_equiv_class_inject"; - - -(**** Defining binary operations upon equivalence classes ****) - - -goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def] - "!!A r. [| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))"; -by (fast_tac rel_cs 1); -qed "congruent2_implies_congruent"; - -val equivA::prems = goalw Equiv.thy [congruent_def] - "[| equiv(A,r); congruent2(r,b); a: A |] ==> \ -\ congruent(r, %x1. UN x2:r^^{a}. b(x1,x2))"; -by (cut_facts_tac (equivA::prems) 1); -by (safe_tac rel_cs); -by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1); -by (assume_tac 1); -by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class, - congruent2_implies_congruent]) 1); -by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]); -by (fast_tac rel_cs 1); -qed "congruent2_implies_congruent_UN"; - -val prems as equivA::_ = goal Equiv.thy - "[| equiv(A,r); congruent2(r,b); a1: A; a2: A |] \ -\ ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b(x1,x2)) = b(a1,a2)"; -by (cut_facts_tac prems 1); -by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class, - congruent2_implies_congruent, - congruent2_implies_congruent_UN]) 1); -qed "UN_equiv_class2"; - -(*type checking*) -val prems = goalw Equiv.thy [quotient_def] - "[| equiv(A,r); congruent2(r,b); \ -\ X1: A/r; X2: A/r; \ -\ !!x1 x2. [| x1: A; x2: A |] ==> b(x1,x2) : B |] \ -\ ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B"; -by (cut_facts_tac prems 1); -by (safe_tac rel_cs); -by (REPEAT (ares_tac (prems@[UN_equiv_class_type, - congruent2_implies_congruent_UN, - congruent2_implies_congruent, quotientI]) 1)); -qed "UN_equiv_class_type2"; - - -(*Suggested by John Harrison -- the two subproofs may be MUCH simpler - than the direct proof*) -val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def] - "[| equiv(A,r); \ -\ !! y z w. [| w: A; : r |] ==> b(y,w) = b(z,w); \ -\ !! y z w. [| w: A; : r |] ==> b(w,y) = b(w,z) \ -\ |] ==> congruent2(r,b)"; -by (cut_facts_tac prems 1); -by (safe_tac rel_cs); -by (rtac trans 1); -by (REPEAT (ares_tac prems 1 - ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1)); -qed "congruent2I"; - -val [equivA,commute,congt] = goal Equiv.thy - "[| equiv(A,r); \ -\ !! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y); \ -\ !! y z w. [| w: A; : r |] ==> b(w,y) = b(w,z) \ -\ |] ==> congruent2(r,b)"; -by (resolve_tac [equivA RS congruent2I] 1); -by (rtac (commute RS trans) 1); -by (rtac (commute RS trans RS sym) 3); -by (rtac sym 5); -by (REPEAT (ares_tac [congt] 1 - ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1)); -qed "congruent2_commuteI"; -